Week 7 Day 2 - Modal Logic
0Today we learned about two modal logic qualifiers - necessary (represented by a square) and possible (represented by a diamond). Something is necessarily true when it must be true in all possible universes, such as Z (integers) having both even and odd numbers in it. Something is possibly true when there is one or more possible universes having that in it, such as me flipping a coin and getting a heads - there's three main outcomes, heads, tails, and landing on the side, and heads could potentially happen, so it is possible that when I flip a coin I will get a heads.
To prove something is necessary usually involves giving some sort of logical proof to show that all outcomes must result in the thing, or by showing it to be true by definition. Triangles are defined as having three sides, so it is necessarily true a triangle has three sides, for example. Or, for example, one can prove that the square root of 2 is irrational by providing a mathematical proof to this effect. This means there is no possible universe where the square root of 2 is rational.
To prove something is possible is easy. Possible is a very weak claim, after all. You just have to show that it is not impossible. So given the hypothesis, show there is no internal contradiction, and you're done. For example, you can't prove it is possible a triangle has four sides (because this would contradict the definition that triangles have three sides), but you can prove it is possible that three-headed aliens exist, since there is no inherent contradiction between "three-headed" and "alien".
Be aware that erasing qualifiers is the most common source of strawmen on the internet. If someone claimed it is possible that three-headed aliens exist, the most common response would be, "Oh yeah? PROVE they exist!" which shows the interlocutor didn't understand what was being claimed. A possibility claim is very weak, but the interlocutor didn't understand this, erased the qualifier, and thought that the first person had said they knew for sure that three-headed aliens exist. And if you give them what they're technically asking for ("There is no contradiction between three-headed and alien, done.") then the other person will usually again fail to understand what just happened, and claim your proof is insufficient, again because they don't understand modal logic.
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